Tautological relations and the \(r\)-spin Witten conjecture. (Relations tautologiques et la conjecture de Witten sur l’espace des structures \(r\)-spin.) (English. French summary) Zbl 1203.53090

Faber, Carel; Shadrin, Sergey; Zvonkine, Dimitri

Ann. Sci. Éc. Norm. Supér. (4) 43, No. 4, 621-658 (2010).

The authors give a simple proof of the fact that the group action, introduced by A. Givental on Gromov-Witten potentials respect tautological relation in the cohomology ring of the moduli space \(\overline{\mathcal{M}}_{g,m}\) of stable pointed maps [Y. P. Lee, C. Teleman].
The authors observe that in any semi-simple GW theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the geometric GW potential coincides with the potential constructed via Givental’s group action. They also show that their results suffice to deduce the statement of a Witten conjecture relating the \(r\)-KdV hierarchy to the intersection theory on the space of \(r\)-spin structures on the stable curves.

Reviewer: Vehbi Emrah Paksoy (Ft. Lauderdale)

MSC:

53D55	Deformation quantization, star products
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45	Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Keywords:

quantization; Frobenius manifolds; Gromov-Witten potential; moduli of curves; r-spin structures; Witten’s conjecture